Some Consequences of the Shadowing Property in Low Dimensions

Abstract

We consider low-dimensional systems with the shadowing property. In dimension two, we show that the shadowing property for a homeomorphism implies the existence of periodic orbits in every $\epsilon$-transitive class, and in contrast we provide an example of a $C^{\infty }$ Kupka-Smale diffeomorphism with the shadowing property exhibiting an aperiodic transitive class. Finally we consider the case of transitive endomorphisms of the circle, and we prove that the $\alpha$-Hölder shadowing property with $\alpha >1/2$ implies that the system is conjugate to an expanding map.

36 pages, 12 figures (revision based on referee's comments. To appear in Ergodic Theory Dynam. Systems)

References in corpus (1)

• Lipschitz shadowing implies structural stability

Cited by in corpus (5)

• Expansivity and Shadowing in Linear Dynamics
• Continuum-wise hyperbolicity
• Stable/unstable holonomies, density of periodic points, and transitivity for continuum-wise hyperbolic homeomorphisms
• Product Anosov diffeomorphisms and the two-sided limit shadowing property
• Continuum-wise hyperbolicity, periodic shadowing, and measures of maximal entropy